Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums

Reny George; Sina Etemad; Ivanka Stamova; Raaid Alubady; Reny George; Sina Etemad; Ivanka Stamova; Raaid Alubady

doi:10.3934/math.2025108

AIMS Mathematics

2025, Volume 10, Issue 2: 2321-2346. doi: 10.3934/math.2025108

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Research article

Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
2.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
3.
Mathematics in Applied Sciences and Engineering Research Group, Scientific Research Center, Al-Ayen University, Nasiriyah 64001, Iraq
4.
Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA
5.
Engineering Technical College, Al-Ayen Iraqi University, Thi-Qar, Iraq
6.
College of Information Technology, University of Babylon, Babel, Iraq

Received: 04 October 2024 Revised: 14 January 2025 Accepted: 21 January 2025 Published: 10 February 2025
MSC : 05A30, 26A33, 39A13, 34A08

The eardrum is one of the most important organs in the body, and disorders such as infection or injury may affect the proper functioning of the eardrum and lead to hearing problems. In this paper, based on a real-world phenomena, we study some mathematical aspects of an abstract fractional $ [\mathtt{p},\mathtt{q}] $-difference equation with initial conditions. Our initial value problem tries to model a vibrating eardrum by using the newly defined fractional Caputo-type $ [\mathtt{p},\mathtt{q}] $-derivatives in two nonlinear single-valued and set-valued structures. We obtain a general form of the solutions in the framework of a $ [\mathtt{p},\mathtt{q}] $-integral equation, and then we investigate the existence and uniqueness properties with the help of fixed points and the end-points of some special $ \mathtt{β} $-$ \mathtt{α} $-contractions and compact mappings. Finally, we simulate this version of the vibrating eardrum model by giving two numerical examples to validate the established theorems.
- set-valued function,
- fixed-point,
- end-point,
- $ [\mathtt{p},\mathtt{q}] $-derivative,
- initial problem,
- vibrating eardrum model
Citation: Reny George, Sina Etemad, Ivanka Stamova, Raaid Alubady. Existence of solutions for $ [\mathtt{p},\mathtt{q}] $-difference initial value problems: application to the $ [\mathtt{p},\mathtt{q}] $-based model of vibrating eardrums[J]. AIMS Mathematics, 2025, 10(2): 2321-2346. doi: 10.3934/math.2025108

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Abstract

The eardrum is one of the most important organs in the body, and disorders such as infection or injury may affect the proper functioning of the eardrum and lead to hearing problems. In this paper, based on a real-world phenomena, we study some mathematical aspects of an abstract fractional $ [\mathtt{p},\mathtt{q}] $-difference equation with initial conditions. Our initial value problem tries to model a vibrating eardrum by using the newly defined fractional Caputo-type $ [\mathtt{p},\mathtt{q}] $-derivatives in two nonlinear single-valued and set-valued structures. We obtain a general form of the solutions in the framework of a $ [\mathtt{p},\mathtt{q}] $-integral equation, and then we investigate the existence and uniqueness properties with the help of fixed points and the end-points of some special $ \mathtt{β} $-$ \mathtt{α} $-contractions and compact mappings. Finally, we simulate this version of the vibrating eardrum model by giving two numerical examples to validate the established theorems.

References

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